The normal and tangential components of stress on a plane inclined at θ to the direction of the force are equal when θ is:
Try it first: pick an option, then read the explanation.
Correct answer: A. 45°
Asked in SSC JE 04/03/2017 (Evening)
Detailed Explanation
Under axial stress σ the two components on an inclined plane are σ cos²θ (normal) and σ sinθ cosθ (tangential). Set them equal.
\[ \sigma\cos^2\theta = \sigma\sin\theta\cos\theta \Rightarrow \tan\theta = 1 \Rightarrow \theta = 45^\circ \]
Both components then equal σ/2.
Why “45°” Is Correct
They are equal at 45°, where each equals σ/2 — and the tangential stress is at its maximum.
Why the Other Options Are Wrong
- B. 30°: At 30°, the components are 0.75σ and 0.43σ (or the reverse).
- C. 60°: At 60°, they are 0.25σ and 0.43σ (or the reverse).
- D. 90°: At 90° one component is zero.
Key Point
Axial stress: σn = τ = σ/2 on the 45° plane.
Key Concepts for Students
- The 45° plane carries the maximum shear σ/2.
- Same answer whichever angle convention is used.
- Ductile failure in tension follows 45° planes.
Practice more: this is question 15 in Principal Stress and Principal Planes MCQs – Part 1. Try the whole set and check every answer instantly.